When 307 Digits Aren’t Enough

A number containing 307 digits — that’s 207 206 more than in a googol — has been factored into an 80-digit number and a 227-digit number. That factorization took 11 months and a century of computing time.

So was it worth it?

This isn’t quite as egg-headed as it sounds, because factoring big numbers is a staple of code-breaking. Cryptologists use large, non-prime numbers to ensure secure communications can only be read by their intended recipient, who holds the factors of those numbers. (That’s an approximate explanation of how this works; If you want the fuller picture, complete with modulus and decryption functions, click here). To crack the code, you must find the factors — which, as you may recall from arithmetic class, involves finding those numbers which divide evenly, without remainders. The factors of 6, for example, are 1, 2, 3 and 6. However, 6 isn’t quite big enough to use for encryption: Those numbers have to keep getting larger to keep pace with increased computing power and improved techniques to hunt for factors. There’s an irregular drumbeat of announcements from code-breaking experts who have harnessed computing centers to break new codes.

The latest such announcement came last week, from a French-German-Japanese consortium. They vanquished the number 21039-1. Previously it was known that the number — I’ll call it N from here on out — could be divided evenly by 5,080,711. But what remained was a 307-digit number that mathematicians hadn’t yet been able to factor, Heise Online reported. This was a logical target because it contained 1,017 binary digits, seven shy of the current encryption standard of 1,024. (Numbers are typically expressed in base-10; in base-2, they look much longer. For instance, 64 in base-10 is 1,000,000 in base-2. You can play with base conversions using this online calculator from Montreal’s Vanier College.)

N is a Mersenne number, meaning it is one fewer than two raised to the power of a prime number — that is, two raised to a number greater than one whose divisors are only itself and one. All such numbers were once believed to be primes, but that theory breaks down by the fifth prime (211-1 = 2,047 = 23*89). French monk Marin Mersenne put forth a slightly more refined, but also incorrect, theory of these numbers and their status as primes, “but still got his name attached to these numbers,” according to this guide to the Mersennes from the University of Tennessee at Martin. Mersenne.org tracks progress in the hunt for new Mersenne primes in the upper stratosphere of big numbers. As of last month, all Mersenne numbers corresponding to prime exponents of two below 15 million have been tested and double-checked.

Such gargantuan numbers make N, with its exponent of 1,039, look puny by comparison. Nonetheless, the press release about N’s factorization was headlined, “A Mighty Number Falls,” and predicted that “the news of this feat will grab the attention of information security experts and may eventually lead to changes in encryption techniques.”

But a Slashdot commenter was skeptical, calling the effort a “stunt” and pointing out that it’s predictable how long such a factorization should take, given a specified computing power and a certain-sized number. Instead, the argument goes, investments should focus on specialized devices for large-scale factoring, such as this one outlined in theory but not yet built. Scientists at Israel’s Weizmann Institute project that, with their theoretical device, “it will be possible to factor 1024-bit integers, and hence to break 1024-bit RSA keys, in 1 year at the cost of a few dozen million US dollars (or significantly less, if several integers are to be factored simultaneously).” That could truly have a seismic impact on cryptography.

About The Numbers

The Wall Street Journal examines numbers in the news, business and politics. Some numbers are flat-out wrong or biased, while others are valid and help us make informed decisions. We tell the stories behind the stats in occasional updates on this blog.